Three-dimensional characterization using the transmission electron microscope (TEM) can reveal complex nanoscale structural and chemical properties. Because the TEM generates two-dimensional images and diffraction patterns, an inversion algorithm is necessary to retrieve the three-dimensional specimen. An inversion algorithm we have developed includes multiple scattering, and thus can provide three-dimensional nanoscale characterization of crystals from TEM diffraction data using artificial-neural-network optimization tools and GPU-accelerated matrix exponentials – we have previously reported retrieving strain and ferroelectric polarization on simulated data [1-2]. Some algorithms reconstruct individual atomic positions [3]; our algorithm retrieves crystal properties and is suitable for larger specimens and structures. Mapping ferroelectric polarization domains and strain state variations in arbitrary geometries as a function of specimen depth with nanometer-scale resolution can enable novel nanoscale analytical insights for a wide range of crystalline materials, including quantifying 3D structures and understanding surface-induced artifacts.

However, our algorithm requires accurately modeling a layered crystalline specimen for Bloch-wave-type calculations. The conventional approach – using isolated-atom scattering factors (IASF) – is fast, but neglects chemical bonding, while directly fitting the Ug structure-factors might result in accurate pattern replication, but with drawbacks for this three-dimensional application – direct-Ug-fitting greatly increases the number of free parameters, and thus likely decreases the precision of each individual parameter while simultaneously complicating analysis [4]. The specifics of this problem enable a third method – density functional theory (DFT) – which provides self-consistent ab-initio structure factors with chemical bonding effects, and has been previously used to precompute key low-order structure factors, comparing well with experiment [4-5]. Here, we use DFT to generate all the structure factors, because the GPAW DFT code, which we have used for mean inner potential calculations, can provide the all-electron density, which can then be processed to yield Ug structure factors [6-7]. With modern hardware, thousands of small-scale DFT simulations can be performed in a reasonable time, enabling DFT integration into our iterative inversion algorithm, which has been improved to be multi-CPU+multi-GPU parallelized.

Figures 1 and 2 show the results of combining DFT-computed self-consistent ab-initio structure factors with our depth-direction parameter retrieval algorithm on simulated data. Simulated SrTiO3 is our test material for simultaneous retrieval of ferroelectric atomic displacements (single-atom property) and oxygen octahedral rotation (multi-atom property); DFT is used both to generate the test data and during the retrieval routine. For perovskites, both of these specific parameters can be of interest for different systems; for other materials, the combination of single-atom and multi-atom parameters could be useful. For this noise-free data, the results cease improving because the DFT simulations use a user-selectable grid spacing; a finer grid can be used, at the cost of computational time.

In this work, we accurately retrieve ferroelectric atomic displacements and perovskite-style octahedral oxygen rotation for SrTiO3 from simulated composite-CBED-type data using ab-initio DFT structure factors [2]. Experimental applications of this technique to both 2D and 3D data will be discussed.